I just did a stainless steel coin door, and it was fun. Here's what you can do:

(front side)
1. polish carriage bolts to a shine using a buffing wheel, OR using attachments for a dremel (you want fast RPMs)
2. regrain the front plate (removing all scratches first)
3. de-rust the coin slot (buy a few gallons of evaporust at local auto-parts store, and Re-use it over and over - PVC pipes are great ways to store it and put long parts in it such as pin-legs.)
4. de-rust the hinge
5. clean up the coin plastics
6. polish the chrome coin slot area
7. de-rust the coin-return lever
8. polish the 4 edges

(back side)
1. clean out lamp sockets
2. (coin mech stuff -- which I haven't messed with yet)
3. removing and cleaning anything attached to the rear of the coin door
4. cleaning the wires leading into the coin door of soda-sugars/dust/gunk

It's up to you how far you take your coin door.

Hope these ideas help.
-mof

To answer this specifically, I'd remove the hinge, set the coindoor inside the pin (I think you can leave the wires attached? Maybe not the ground wire) and leave the hinge part overnight in an evaporust solution, then clean and polish it and reattach it.

If you are asking, can you use a product and a rag to clean it? You could, but it's not the way I'd approach it. I'd want to get in every crack and surface, and an overnight soak is best for that.

-mof

Mother's mag & aluminum polish
tin foil